Over the years, there has been much interest in non-Hermitian Hamiltonians with a real spectrum, most of them describing one-dimensional systems with unbroken PT-invariance [1]}, [2]}, [3]} or, more generally, endowed with a pseudo-Hermiticity property, i.e., such that \(\eta H \eta ^{-1} = H^{\dagger }\)  with \(\eta \)  a Hermitian invertible operator [4]}, [5]}. For such systems, the Hilbert space is described is terms of a biorthogonal basis consisting of the eigenstates \(\Psi _n\)  and \(\tilde{\Psi }_n\)  of \(H\)  and \(H^{\dagger }\) , respectively.
